### 2nd Putnam 1939

**Problem A2**

Let C be the curve y = x^{3} (where x takes all real values). The tangent at A meets the curve again at B. Prove that the gradient at B is 4 times the gradient at A.

**Solution**

Trivial. [Take the point as (a,a^{3}). Write down the equation of the tangent. Write down its point of intersection with the curve: (x^{3} - a^{3}) = 3a^{2}(x - a). We know this has a repeated root x = a. The sum of the roots is zero, so the third root is x = - 2a. Finally, 3(-2a)^{2} = 4 times 3 a^{2}.]

2nd Putnam 1939

© John Scholes

jscholes@kalva.demon.co.uk

4 Sep 1999